Although Huck has several moral/emotional transformations in the novel, I believe you're referring to the moment in chapter 23, when Huck realizes that Jim cares for his family just as much as a white man would. Huck awakes to find Jim crying, and describes his reaction:

I went to sleep, and Jim didn't call me when it was my turn. He often done that. When I waked up just at daybreak he was sitting there with his head down betwixt his knees, moaning and mourning to himself. I didn't take notice nor let on. I knowed what it was about. He was thinking about his wife and his children, away up yonder, and he was low and homesick; because he hadn't ever been away from home before in his life; and I do believe he cared just as much for his people as white folks does for their'n. It don't seem natural, but I reckon it's so.

First, we see that Jim cares for Huck, going out of his way to take longer shifts on watch so Huck can rest. He is a true friend, which Huck recognizes on some level, although he isn't able to articulate it at this point. But, we do see a shift from earlier chapters, where Huck is shocked by Jim's desire to rescue his children from slavery. In chapter 16, Huck considers Jim's children property of their slaveowner, & doesn't recognize the deep connection between Jim and his family. Yet we see the change in Huck here. He slowly understands that Jim does indeed have the same emotions and relationships as all humans, although he remarks that it "doesn't seem natural". This is an important moment for Huck, and it sets the stage for his decision to "go to hell" instead of turning Jim in later.